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<h1>Section 5.2.5: Mixed strategies for matrix games (LP formulation)</h1>
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<pre class="codeinput">
<span class="comment">% Boyd &amp; Vandenberghe, "Convex Optimization"</span>
<span class="comment">% Jo&euml;lle Skaf - 08/24/05</span>
<span class="comment">%</span>
<span class="comment">% Player 1 wishes to choose u to minimize his expected payoff u'Pv, while</span>
<span class="comment">% player 2 wishes to choose v to maximize u'Pv, where P is the payoff</span>
<span class="comment">% matrix, u and v are the probability distributions of the choices of each</span>
<span class="comment">% player (i.e. u&gt;=0, v&gt;=0, sum(u_i)=1, sum(v_i)=1)</span>
<span class="comment">% LP formulation:   minimize    t</span>
<span class="comment">%                       s.t.    u &gt;=0 , sum(u) = 1, P'*u &lt;= t*1</span>
<span class="comment">%                   maximize    t</span>
<span class="comment">%                       s.t.    v &gt;=0 , sum(v) = 1, P*v &gt;= t*1</span>

<span class="comment">% Input data</span>
randn(<span class="string">'state'</span>,0);
n = 12;
m = 12;
P = randn(n,m);

<span class="comment">% Optimal strategy for Player 1</span>
fprintf(1,<span class="string">'Computing the optimal strategy for player 1 ... '</span>);

cvx_begin
    variables <span class="string">u(n)</span> <span class="string">t1</span>
    minimize ( t1 )
    u &gt;= 0;
    ones(1,n)*u == 1;
    P'*u &lt;= t1*ones(m,1);
cvx_end

fprintf(1,<span class="string">'Done! \n'</span>);

<span class="comment">% Optimal strategy for Player 2</span>
fprintf(1,<span class="string">'Computing the optimal strategy for player 2 ... '</span>);

cvx_begin
    variables <span class="string">v(m)</span> <span class="string">t2</span>
    maximize ( t2 )
    v &gt;= 0;
    ones(1,m)*v == 1;
    P*v &gt;= t2*ones(n,1);
cvx_end

fprintf(1,<span class="string">'Done! \n'</span>);

<span class="comment">% Displaying results</span>
disp(<span class="string">'------------------------------------------------------------------------'</span>);
disp(<span class="string">'The optimal strategies for players 1 and 2 are respectively: '</span>);
disp([u v]);
disp(<span class="string">'The expected payoffs for player 1 and player 2 respectively are: '</span>);
[t1 t2]
disp(<span class="string">'They are equal as expected!'</span>);
</pre>
<a id="output"></a>
<pre class="codeoutput">
Computing the optimal strategy for player 1 ...  
Calling Mosek 9.1.9: 25 variables, 13 equality constraints
   For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86

Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : LO (linear optimization problem)
  Constraints            : 13              
  Cones                  : 0               
  Scalar variables       : 25              
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 1                 time                   : 0.00            
Lin. dep.  - tries                  : 1                 time                   : 0.00            
Lin. dep.  - number                 : 0               
Presolve terminated. Time: 0.00    
Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : LO (linear optimization problem)
  Constraints            : 13              
  Cones                  : 0               
  Scalar variables       : 25              
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 13
Optimizer  - Cones                  : 0
Optimizer  - Scalar variables       : 26                conic                  : 0               
Optimizer  - Semi-definite variables: 0                 scalarized             : 0               
Factor     - setup time             : 0.00              dense det. time        : 0.00            
Factor     - ML order time          : 0.00              GP order time          : 0.00            
Factor     - nonzeros before factor : 91                after factor           : 91              
Factor     - dense dim.             : 0                 flops                  : 3.34e+03        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   2.2e+01  2.0e+00  2.0e+00  0.00e+00   0.000000000e+00   0.000000000e+00   4.0e+00  0.00  
1   5.9e+00  5.4e-01  5.4e-01  2.94e+00   2.491088385e-02   -4.511300985e-02  1.1e+00  0.01  
2   2.4e+00  2.1e-01  2.1e-01  1.65e+00   2.359713864e-02   -2.703348050e-03  4.3e-01  0.01  
3   3.6e-01  3.3e-02  3.3e-02  1.27e+00   4.372219258e-02   4.008718888e-02   6.6e-02  0.01  
4   4.5e-02  4.1e-03  4.1e-03  1.04e+00   4.432272369e-02   4.387577779e-02   8.2e-03  0.01  
5   1.3e-03  1.1e-04  1.1e-04  1.02e+00   4.484616797e-02   4.483360596e-02   2.3e-04  0.01  
6   3.3e-07  3.0e-08  3.0e-08  1.00e+00   4.484042333e-02   4.484042011e-02   6.0e-08  0.01  
Basis identification started.
Primal basis identification phase started.
Primal basis identification phase terminated. Time: 0.00
Dual basis identification phase started.
Dual basis identification phase terminated. Time: 0.00
Basis identification terminated. Time: 0.00
Optimizer terminated. Time: 0.01    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: 4.4840423334e-02    nrm: 1e+00    Viol.  con: 4e-08    var: 0e+00  
  Dual.    obj: 4.4840420109e-02    nrm: 7e-01    Viol.  con: 0e+00    var: 1e-09  

Basic solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: 4.4840422221e-02    nrm: 1e+00    Viol.  con: 6e-17    var: 0e+00  
  Dual.    obj: 4.4840420109e-02    nrm: 7e-01    Viol.  con: 0e+00    var: 6e-09  
Optimizer summary
  Optimizer                 -                        time: 0.01    
    Interior-point          - iterations : 6         time: 0.01    
      Basis identification  -                        time: 0.00    
        Primal              - iterations : 0         time: 0.00    
        Dual                - iterations : 0         time: 0.00    
        Clean primal        - iterations : 0         time: 0.00    
        Clean dual          - iterations : 0         time: 0.00    
    Simplex                 -                        time: 0.00    
      Primal simplex        - iterations : 0         time: 0.00    
      Dual simplex          - iterations : 0         time: 0.00    
    Mixed integer           - relaxations: 0         time: 0.00    

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -0.0448404
 
Done! 
Computing the optimal strategy for player 2 ...  
Calling Mosek 9.1.9: 25 variables, 13 equality constraints
   For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86

Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : LO (linear optimization problem)
  Constraints            : 13              
  Cones                  : 0               
  Scalar variables       : 25              
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 1                 time                   : 0.00            
Lin. dep.  - tries                  : 1                 time                   : 0.00            
Lin. dep.  - number                 : 0               
Presolve terminated. Time: 0.00    
Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : LO (linear optimization problem)
  Constraints            : 13              
  Cones                  : 0               
  Scalar variables       : 25              
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 13
Optimizer  - Cones                  : 0
Optimizer  - Scalar variables       : 26                conic                  : 0               
Optimizer  - Semi-definite variables: 0                 scalarized             : 0               
Factor     - setup time             : 0.00              dense det. time        : 0.00            
Factor     - ML order time          : 0.00              GP order time          : 0.00            
Factor     - nonzeros before factor : 91                after factor           : 91              
Factor     - dense dim.             : 0                 flops                  : 3.34e+03        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   2.2e+01  2.0e+00  2.0e+00  0.00e+00   0.000000000e+00   0.000000000e+00   4.0e+00  0.00  
1   7.8e+00  7.1e-01  7.1e-01  3.11e+00   -4.221775224e-02  -6.367209421e-02  1.4e+00  0.01  
2   2.2e+00  2.0e-01  2.0e-01  1.62e+00   -5.045269958e-02  -7.643306967e-02  3.9e-01  0.01  
3   2.9e-01  2.6e-02  2.6e-02  1.31e+00   -4.794188101e-02  -5.120950932e-02  5.3e-02  0.01  
4   2.9e-02  2.6e-03  2.6e-03  1.03e+00   -4.477461238e-02  -4.509672836e-02  5.3e-03  0.01  
5   6.2e-04  5.6e-05  5.6e-05  1.01e+00   -4.485260682e-02  -4.485940940e-02  1.1e-04  0.01  
6   1.6e-07  1.5e-08  1.5e-08  1.00e+00   -4.484042510e-02  -4.484042684e-02  2.9e-08  0.01  
Basis identification started.
Primal basis identification phase started.
Primal basis identification phase terminated. Time: 0.00
Dual basis identification phase started.
Dual basis identification phase terminated. Time: 0.00
Basis identification terminated. Time: 0.00
Optimizer terminated. Time: 0.01    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: -4.4840425103e-02   nrm: 1e+00    Viol.  con: 2e-08    var: 0e+00  
  Dual.    obj: -4.4840426844e-02   nrm: 5e-01    Viol.  con: 0e+00    var: 1e-09  

Basic solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: -4.4840422221e-02   nrm: 1e+00    Viol.  con: 2e-16    var: 0e+00  
  Dual.    obj: -4.4840426844e-02   nrm: 5e-01    Viol.  con: 0e+00    var: 3e-09  
Optimizer summary
  Optimizer                 -                        time: 0.01    
    Interior-point          - iterations : 6         time: 0.01    
      Basis identification  -                        time: 0.00    
        Primal              - iterations : 0         time: 0.00    
        Dual                - iterations : 0         time: 0.00    
        Clean primal        - iterations : 0         time: 0.00    
        Clean dual          - iterations : 0         time: 0.00    
    Simplex                 -                        time: 0.00    
      Primal simplex        - iterations : 0         time: 0.00    
      Dual simplex          - iterations : 0         time: 0.00    
    Mixed integer           - relaxations: 0         time: 0.00    

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -0.0448404
 
Done! 
------------------------------------------------------------------------
The optimal strategies for players 1 and 2 are respectively: 
    0.2695    0.0686
   -0.0000    0.1619
    0.0973    0.0000
    0.1573    0.2000
    0.1145   -0.0000
    0.0434    0.1545
    0.0000    0.1146
    0.0000   -0.0000
    0.2511    0.1030
    0.0670   -0.0000
   -0.0000    0.0000
    0.0000    0.1974

The expected payoffs for player 1 and player 2 respectively are: 

ans =

   -0.0448   -0.0448

They are equal as expected!
</pre>
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